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A stochastic epidemic model with memory of the last infection and waning immunity

Hélène Guérin, Arsene Brice Zotsa-Ngoufack

TL;DR

This work extends stochastic SIS-like epidemic models to incorporate memory of the last infection through a trait-age structure, yielding a nonlinear PDE in the large-population limit. It proves a functional law of large numbers and propagation of chaos, linking the microscopic stochastic dynamics to a deterministic density $u_t(a,\theta)$ evolving under a PDE with a population force of infection $\mathfrak{F}(t)$ and a time-dependent susceptibility $\mathfrak{S}(t,\theta)$. A general endemicity threshold is derived via a spectral analysis of a kernel operator, with conditions for the existence and uniqueness of endemic equilibria, and local stability results in a memoryless setting. The paper also applies the threshold framework to vaccination policies, deriving explicit endemicity criteria for a single-shot policy and a renewal-vaccination policy, and demonstrates that familiar results are recovered in the appropriate limits. These findings illuminate how memory and vaccination strategies shape long-term disease persistence and provide tractable criteria for assessing endemic risk in memory-aware epidemiological models.

Abstract

We adapt the article of Forien, Pang, Pardoux and Zotsa: Arxiv preprint Arxiv2210.04667(2022), on epidemic models with varying infectivity and waning immunity, to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.

A stochastic epidemic model with memory of the last infection and waning immunity

TL;DR

This work extends stochastic SIS-like epidemic models to incorporate memory of the last infection through a trait-age structure, yielding a nonlinear PDE in the large-population limit. It proves a functional law of large numbers and propagation of chaos, linking the microscopic stochastic dynamics to a deterministic density evolving under a PDE with a population force of infection and a time-dependent susceptibility . A general endemicity threshold is derived via a spectral analysis of a kernel operator, with conditions for the existence and uniqueness of endemic equilibria, and local stability results in a memoryless setting. The paper also applies the threshold framework to vaccination policies, deriving explicit endemicity criteria for a single-shot policy and a renewal-vaccination policy, and demonstrates that familiar results are recovered in the appropriate limits. These findings illuminate how memory and vaccination strategies shape long-term disease persistence and provide tractable criteria for assessing endemic risk in memory-aware epidemiological models.

Abstract

We adapt the article of Forien, Pang, Pardoux and Zotsa: Arxiv preprint Arxiv2210.04667(2022), on epidemic models with varying infectivity and waning immunity, to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.
Paper Structure (16 sections, 18 theorems, 131 equations)

This paper contains 16 sections, 18 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. A stochastic parametric model with memory
  4. Main results
  5. Functional law of Large Numbers
  6. The deterministic limit
  7. Propagation of chaos
  8. Long time behavior
  9. Existence of a stationary measure
  10. About the local stability of endemic equilibria
  11. Local stability of endemicity for a SIS-type model
  12. Application to models incorporating a vaccination policy
  13. One shot of vaccination after an infection
  14. A renewal process of vaccination
  15. Absolute continuity of the solution
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $\lambda$ and $\gamma$ be non-negative measurable bounded functions respectively by $\lambda_*$ and $1$. Under Assumption Hyp:Kernel, as $N\to\infty,\,\mu ^N$ converges in law to a measure $\mu \in\mathbb{D}(\mathbb{R}_+;\mathcal{P}(\mathbb{R}_+\times\Theta))$, which is the unique solution to for any measurable bounded function $f$ on $\mathbb{R}_+\times\mathbb{R}_+\times\Theta$, of class $\m

Theorems & Definitions (42)

  • Theorem 3.1
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 3.8
  • Remark 4.1
  • Proposition 4.2
  • ...and 32 more